Abstract:

The theory on the interaction of a cumulus cloud ensemble with the large scale environment originally proposed by Arakawa (1971) and Arakawa and Schubert (1973) is presented. According to this theory the large scale environment is divided into the subcloud mixed layer and the region above. Time changes of the large scale environment are given by the heat and moisture budget equations for each region and also by an equation which predicts the height of the mixed layer. Above the mixed layer the cumulus convection affects the large scale temperature and moisture fields through detrainment and cumulus induced subsidence. The detraining mass is saturated air containing liquid water, which evaporates in the environment. Detrainment thus causes large scale cooling and moistening. Cumulus induced subsidence typically causes large scale warming and drying. The temperature and moisture fields in the mixed layer are not directly affected by the presence of cumulus convection but the top of the mixed layer is affected by cumulus induced subsidence. The problem of parameterization turns out to be the determination of the vertical profiles of three quantities : the total vertical mass flux in the clouds, M[subscript c] ; the total detrainment of mass from the clouds, D; and the detraining cloud liquid water content, ℓ̂. The cumulus ensemble is spectrally divided into sub-ensembles according to fractional entrainment rate [lambda]. The sub-ensemble budget equations for moist static energy and total water content are given. Solution of these equations gives the temperature excess, water vapor excess, and liquid water content of the clouds. Clouds with smaller fractional entrainment rates retain their buoyancy longer and are thus deeper than clouds with large fractional entrainment rates . Solution of the sub-ensemble budget equations reduces the problem of parameterization to the determination of M[subscript B](lambda) d lambda, the vertical mass flux at the top of the mixed layer due to all clouds with fractional entrainment rates between [lambda] and [lambda + d lambda]. An integral equation for the mass flux distribution function M[subscript B](lambda) is derived. This integral equation, which forms the heart of the theory, describes how a cumulus ensemble is forced by large scale processes. The kernel of this integral equation can be divided Into three parts : detrainment kernel, vertical mass flux kernel, and mixed layer kernel. These kernels describe how different members of the cloud ensemble influence one another through modification of the environment. The forcing term in this Integral equation can be divided into two parts : cloud layer forcing and mixed layer forcing. These forcing functions describe how large scale advection, radiation, and surface turbulent fluxes can force convection. Numerical methods for solving the integral equation are discussed. An application of the theory is then made to Marshall Islands data. The mass flux distribution function found in a region of time average large scale ascent (Intertropical Convergence Zone) is remarkably different from that found in a region of time average large scale descent (Trade Wind Zone). In the Intertropical Convergence Zone most of the ensemble vertical mass flux occurs in deep convection, while in the Trade Wind Zone most of the ensemble vertical mass flux occurs in shallow convection.